The proper structure for this scenario is a SubClass / Inheritance model, and is nearly identical to the concept I proposed in this answer: Heterogeneous ordered list of value.

The model proposed in this question is actually quite close in that the Animal entity contains the type (i.e. race) and the properties that are common across all types. However, there are two minor changes that are needed:

1. Remove the Cat_ID and Dog_ID fields from their respective entities:

   The key concept here is that everything is an Animal, regardless of race: Cat, Dog, Elephant, and so on. Given that starting point, any particular race of Animal doesn't truly need a separate identifier since:

   1. the Animal_ID is unique
   2. the Cat, Dog, and any additional race entities added in the future do not, by themselves, fully represent any particular Animal; they only have meaning when used in combination with the information contained in the parent entity, Animal.

   Hence, the Animal_ID property in the Cat, Dog, etc entities is both the PK and the FK back to the Animal entity.

2. Differentiate between types of breed:

   Just because two properties share the same name does not necessarily mean that those properties are the same, even if the name being the same implies such a relationship. In this case, what you really have is actually CatBreed and DogBreed as seperate "types"

Initial Notes

1. The SQL is specific to Microsoft SQL Server (i.e. is T-SQL). Meaning, be careful about datatypes as they are not the same across all RDBMS's. For example, I am using VARCHAR but if you need to store anything outside of the standard ASCII set, you should really use NVARCHAR.
2. The ID fields of the "type" tables (Race, CatBreed, and DogBreed) are not auto-incrementing (i.e. IDENTITY in terms of T-SQL) because they are application constants (i.e. they are part of the application) that are static lookup values in the database and are represented as enums in C# (or other languages). If values are added, they are added in controlled situations. I reserve the use of auto-increment fields for user data that comes in via the application.
3. The naming convention I use is to name each subclass table starting with the main class name followed by the subclass name. This helps organize the tables as well as indicates clearly (without looking at the FKs) the relationship of the subclass table to the main entity table.
4. Please see "Final Edit" section at the end for a note regarding Views.

"Breed" as "Race"-Specific Approach

This first set of tables are the lookup / types tables:

CREATE TABLE Race
(
  RaceID INT NOT NULL PRIMARY KEY
  RaceName VARCHAR(50) NOT NULL
);

CREATE TABLE CatBreed
(
  CatBreedID INT NOT NULL PRIMARY KEY,
  BreedName VARCHAR(50),
  CatBreedAttribute1 INT,
  CatBreedAttribute2 VARCHAR(10)
  -- other "CatBreed"-specific properties as needed
);

CREATE TABLE DogBreed
(
  DogBreedID INT NOT NULL PRIMARY KEY,
  BreedName VARCHAR(50),
  DogBreedAttribute1 TINYINT
  -- other "DogBreed"-specific properties as needed
);

This second listing is the main "Animal" entity:

CREATE TABLE Animal
(
  AnimalID INT NOT NULL IDENTITY(1, 1) PRIMARY KEY,
  RaceID INT NOT NULL, -- FK to Race
  Name VARCHAR(50)
  -- other "Animal" properties that are shared across "Race" types
);

ALTER TABLE Animal
  ADD CONSTRAINT [FK_Animal_Race]
  FOREIGN KEY (RaceID)
  REFERENCES Race (RaceID);

This third set of tables are the complimentary sub-class entities that complete the definition of each Race of Animal:

CREATE TABLE AnimalCat
(
  AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
  CatBreedID INT NOT NULL, -- FK to CatBreed
  HairColor VARCHAR(50) NOT NULL
  -- other "Cat"-specific properties as needed
);

ALTER TABLE AnimalCat
  ADD CONSTRAINT [FK_AnimalCat_CatBreed]
  FOREIGN KEY (CatBreedID)
  REFERENCES CatBreed (CatBreedID);

ALTER TABLE AnimalCat
  ADD CONSTRAINT [FK_AnimalCat_Animal]
  FOREIGN KEY (AnimalID)
  REFERENCES Animal (AnimalID);


CREATE TABLE AnimalDog
(
  AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
  DogBreedID INT NOT NULL, -- FK to DogBreed
  HairColor VARCHAR(50) NOT NULL
  -- other "Dog"-specific properties as needed
);

ALTER TABLE AnimalDog
  ADD CONSTRAINT [FK_AnimalDog_DogBreed]
  FOREIGN KEY (DogBreedID)
  REFERENCES DogBreed (DogBreedID);

ALTER TABLE AnimalDog
  ADD CONSTRAINT [FK_AnimalDog_Animal]
  FOREIGN KEY (AnimalID)
  REFERENCES Animal (AnimalID);

The model using a shared breed type is shown after the "Additional Notes" section.

Additional Notes

1. The concept of breed seems to be a focal point for confusion. It was suggested by jcolebrand (in a comment on the question) that breed is a property shared across the different races, and the other two answers have it integrated as such in their models. This is a mistake, however, because the values for breed are not shared across the different values of race. Yes, I am aware that the two other proposed models attempt to solve this issue by making race a parent of breed. While that technically solves the relationship issue, it doesn't help solve the overall modeling question of what to do about non-common properties, nor how to handle a race that does not have a breed. But, in the case that such a property were guaranteed to exist across all Animals, I will include an option for that as well (below).
2. The models proposed by vijayp and DavidN (which seem to be identical) do not work because:
   1. They either
      1. do not allow for non-common properties to be stored (at least not for individual instances of any Animal), or
      2. require that all properties for all races be stored in the Animal entity which is a very flat (and nearly non-relational) way of representing this data. Yes, people do this all of the time, but it means having many NULL fields per row for the properties that are not meant for that particular race AND knowing which fields per row are associated with the particular race of that record.
   2. They do not allow for adding a race of Animal in the future that does not have breed as a property. And even if ALL Animals have a breed, that wouldn't change the structure due to what has been previously noted about breed: that breed is dependent on the race (i.e. breed for Cat is not the same thing as breed for Dog).

"Breed" as Common- / Shared- Property Approach

Please note:

1. The SQL below can be run in the same database as the model presented above:

   1. The Race table is the same
   2. The Breed table is new
   3. The three Animal tables have been appended with a 2
2. Even with Breed being a now common property, it does not seem right not to have Race noted in the main/parent entity (even if it is technically relationally correct). So, both RaceID and BreedID are represented in Animal2. In order to prevent a mismatch between the RaceID noted in Animal2 and a BreedID that is for a different RaceID, I have added a FK on both RaceID, BreedID that references a UNIQUE CONSTRAINT of those fields in the Breed table. I usually despise pointing a FK to a UNIQUE CONSTRAINT, but here is one of the few valid reasons to do so. A UNIQUE CONSTRAINT is logically an "Alternate Key", which makes it valid for this use. Please also note that the Breed table still has a PK on just BreedID.
   1. The reason for not going with just a PK on the combined fields and no UNIQUE CONSTRAINT is that it would allow for the same BreedID to be repeated across different values of RaceID.
   2. The reason for not switching which the PK and UNIQUE CONSTRAINT around is that this might not be the only usage of BreedID, so it should still be possible to reference a specific value of Breed without having the RaceID available.
3. While the following model does work, it has two potential flaws regarding the concept of a shared Breed (and are why I prefer the Race-specific Breed tables).
   1. There is an implicit assumption that ALL values of Breed have the same properties. There is no easy way in this model to have disparate properties between Dog "breeds" and Elephant "breeds". However, there still is a way to do this, which is noted in the "Final Edit" section.
   2. There is no way to share a Breed across more than one race. I am not sure if that is desirable to do (or maybe not in the concept of animals but possibly in other situations that would be using this type of model), but it is not possible here.

CREATE TABLE Race
(
  RaceID INT NOT NULL PRIMARY KEY,
  RaceName VARCHAR(50) NOT NULL
);

CREATE TABLE Breed
(
  BreedID INT NOT NULL PRIMARY KEY,
  RaceID INT NOT NULL, -- FK to Race
  BreedName VARCHAR(50)
);

ALTER TABLE Breed
  ADD CONSTRAINT [UQ_Breed]
  UNIQUE (RaceID, BreedID);

ALTER TABLE Breed
  ADD CONSTRAINT [FK_Breed_Race]
  FOREIGN KEY (RaceID)
  REFERENCES Race (RaceID);

CREATE TABLE Animal2
(
  AnimalID INT NOT NULL IDENTITY(1, 1) PRIMARY KEY,
  RaceID INT NOT NULL, -- FK to Race, FK to Breed
  BreedID INT NOT NULL, -- FK to Breed
  Name VARCHAR(50)
  -- other properties common to all "Animal" types
);

ALTER TABLE Animal2
  ADD CONSTRAINT [FK_Animal2_Race]
  FOREIGN KEY (RaceID)
  REFERENCES Race (RaceID);

-- This FK points to the UNIQUE CONSTRAINT on Breed, _not_ to the PK!
ALTER TABLE Animal2
  ADD CONSTRAINT [FK_Animal2_Breed]
  FOREIGN KEY (RaceID, BreedID)
  REFERENCES Breed (RaceID, BreedID);


CREATE TABLE AnimalCat2
(
  AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
  HairColor VARCHAR(50) NOT NULL
);

ALTER TABLE AnimalCat2
  ADD CONSTRAINT [FK_AnimalCat2_Animal2]
  FOREIGN KEY (AnimalID)
  REFERENCES Animal2 (AnimalID);

CREATE TABLE AnimalDog2
(
  AnimalID INT NOT NULL PRIMARY KEY,
  HairColor VARCHAR(50) NOT NULL
);

ALTER TABLE AnimalDog2
  ADD CONSTRAINT [FK_AnimalDog2_Animal2]
  FOREIGN KEY (AnimalID)
  REFERENCES Animal2 (AnimalID);

Final Edit (hopefully ;-)

1. Regarding the possibility (and then difficulty) of handling disparate properties between types of Breed, it is possible to employ the same subclass / inheritance concept but with Breed as the main entity. In this setup the Breed table would have the properties common to all types of Breed (just like the Animal table) and RaceID would represent the type of Breed (same as it does in the Animal table). Then you would have subclass tables such as BreedCat, BreedDog, and so on. For smaller projects this might be considered "over-engineering", but it is being mentioned as an option for situations that would benefit from it.

2. For both approaches, it sometimes helps to create Views as short-cuts to the full entities. For example, consider:

   CREATE VIEW Cats AS
      SELECT  an.AnimalID,
              an.RaceID,
              an.Name,
              -- other "Animal" properties that are shared across "Race" types
              cat.CatBreedID,
              cat.HairColor
              -- other "Cat"-specific properties as needed
      FROM    Animal an
      INNER JOIN  AnimalCat cat
              ON  cat.AnimalID = an.AnimalID
      -- maybe add in JOIN(s) and field(s) for "Race" and/or "Breed"
   

3. While not part of the logical entities, it is fairly common to have audit fields in the tables to at least get a sense of when the records are being inserted and updated. So in practical terms:
   1. A CreatedDate field would be added to the Animal table. This field is not needed in any of the subclass tables (e.g. AnimalCat) as the rows being inserted for both tables should be done at the same time within a transaction.
   2. A LastModifiedDate field would be added to the Animal table and all subclass tables. This field gets updated only if that particular table is updated: if an update occurs in AnimalCat but not in Animal for a particular AnimalID, then only the LastModifiedDate field in AnimalCat would be set.

It is great that you are taking the time to understand, classify and model the data you are dealing with since, from my personal experiencie, all this makes the whole development process easier and very flexible for future changes. And I am quite sure that you are also aware of this already.

Preliminary data model and assumed business rules

I defined a list of business rules that I have assumed after reading your question and examining closely your diagrams, in order to describe my understanging of your specifications. After defining such list, I derived an IDEF1X^[1] data model that I decided to upload as a .PDF document in an external platform (Dropbox), since due to its format this data model does not fit well in an embedded image. These two instruments are going to be useful as references for some important points that I enumerate below in the section entitled Aspects to resolve in order to keep moving forward.

First, here is the…

• Organizations and Profiles Preliminary Data Model.

Since it is only that, preliminary, consider it as an means helping us to acomplish the desired final data model.

Assumed business rules

Said preliminary data model was derived from a collection of business rules (inferred from your question) that I will enumerate as follows:

Organizations and profiles

Note that Profile is currently understood as a synonym for Person.

• An Organization is a friend of one-to-many Profiles.
• An Organization is a friend of one-to-many Organizations.
• An Organization is a member of one-to-many Organizations.
• A Profile is a member of one-to-manyOrganizations.
• A Profile is a friend of one-to-many Profiles.
• A Profile is a member of one-to-many Profiles.

Locations and addresses

• An Organization owns one-to-many Locations.

• A Location is classified by one-to-many LocationTypes (only one at a given point in time).

• A Location may have one-to-many Addresses (one Physical, one for Shipping, one for Billing, or one that serves all said purposes, or one that combines two purposes and another that serves only one of them).

• An Address may be kept by one-to-many Profiles or, put another way, a Profile keeps one-to-many Addresses.

• A specific Address may be used by one-to-many Profiles (serving as Physical for one Profile, being used for Billing by a different one, etc.). So, an Address works in a similar way for Locations and Profiles.

  • Thus, an individual Address may be, at the same time, of type Physical, Shipping and Billing.

Locations and roles

• A Location opens one-to-many Roles.
• A Role may be carried out in one-to-many Locations.
• A Profile (once it has been set as Member of an Organization) may carry out one-to-many Roles, in one-to-many Locations (but only one specific Role in each Location at a particular point in time, i.e., never two or more Roles at the same time).

Aspects to resolve in order to keep moving forward

In order to keep advancing in the resolution of your data model, here is a list of relevant points that, once we work them out, are going to help us to reach this goal:

1. I have assumed that the term Profile in your context has a similar (or the same) meaning as that of Person, but it could be a bit different. In this way, would you say that, in your scenario, the entities Organization and Person are subtypes of Profile?

2. Can a Profile (or Person) own one-to-many EmailAddresses, or is a Profile (or Person) fixed to exactly one EmailAddress?

3. Would you like to provide the possibility for an Organization to be contacted via Telephone and Email, or you want to restrict that to be possible only for a Profile (or Person)?

4. I assume that a Location is fixed to exactly one Address of the type Physical, is this correct?

5. Is it possible for a Location to be shared by one-to-many different Organizations or, otherwise, a Location can be owned by only one Organization?

6. You have stated via comments that the fact of being a Member and a Friend is the same. As you can see in my proposed preliminary data model, I followed you original specifications and depicted all the possible combinations of membership and friendship between Organization and Profile (or Person) in different entities since I think that it can be helpful in the effort of defining the best possible structure for that part of your scenario. In this sense:

   • I assume that the the statement an Organization is a Member of another Organization has different effects than the statement a Profile (or Person) is a Member of an Organization regards the entity called Location.
   • As you can see in the data model, I think that the Role of Owner is only valid for an Organization and, to me, the valid Roles for a Profile (or Person), inside a Location are Admin and Member. What do you think about all this? Since you are in direct contact with the business rules that apply to your situation, you need to tell me if my assumptions are correct.

7. Can a Profile (or Person) play different Roles inside the same Location? i.e., can a Person be, at the same time, the Admin and also a Member of the same Location? What are the rules in this regard?

8. I think that the same Profile (or Person) can play different Roles in different Locations. For instance: A specific Profile (or Person) is the “Admin” in Location “1”, and this same Profile (or Person) is a “Member” in Location “2”, at the same time. Am I right?

9. Is it possible for a particular Location to have different LocationTypes at the same time, or is an individual Location fixed to hold exactly one LocationType?

10. Does the attribute Organization.Website represent the website address of a particular organization, such as “dba.stackexchange.com”?

11. If Profile “1” (understood as Person) is a Member (or Friend) of Profile “2”, is it possible for Profile “1” to carry out a Role in a Location owned by Profile “2”? I consider that such scenarios are only valid for the relationships between an Organization and a Member Person so, what do you think?

12. In a similar way, if Organization “1” is a Member (or Friend) of Organization “2”, is it possible for Organization “1” to carry out a Role in a Location owned by Organization “2”? Again, I think that this kind of scenarios are only valid for the relationships between an Organization and a Member Person, is this correct?

13. In this regard —while I am writing this questions— I think that it would be reasonable to say that there are only three different kinds relationships involving Organizations and Persons, and we can define:

    • (a) The relationship between an Organization and a Person as “Membership”.
    • (b) The relationship between a Person and another different Person as “Friendhip”.
    • (c) We have yet to find a meaningful name in order to describe the relationship between an individual Organization and another different Organization.
    • So, let me know what you think about (a), (b) and (c).

14. Is it possible for an Organization to be a Friend (or a Member) of one-to-many different Organizations at the same time? Or it is only possible for an Organization to have only a relationship with exclusively one different Organization?

Successive data model depicting the first advance

In attention to your responses and resolutions to the pending aspects that I have listed above, I have created the following…

• Organizations and Profiles Preliminary Data Model — First Advance.

Although I do not feel quite comfortable with it yet, this new data model expresses the following business rules:

• A Profile is either an Organization or a Person.^[2]
• A Profile may be the offering friend of one-to-many FriendProfiles, and a Profile may be the accepting friend of one-to-many FriendProfiles.^[3]
• A Location may consist of one-to-many Locations.^[4]

Answers to your subsequent specific comments

It's really interesting for me to note/compound the separation of concerns [e.g. LocationAddress and ProfileAddress] - for I obviously wanted to rush in and hold them all without the correct relations [funnily, It didn't feel right with my original ERD].

Yes, that is a good comparison, although I would not call it separation of concerns (which is, certainly, a fundamental principle in application programming and design), since this term commonly pertains to the application development stage and we currently find ourselves in the stage of understanding the data and designing its logical structure.

From my personal experience, I consider that this phase has to do with putting the significant things into their whole context, it has to do with seeing the associations that exist between the different entities that are of relevance in the particular scenario of interest, and then depicting these things in a data model. In the specific case on which you are commenting about, the Address entity may have different kinds of connections with other entities, one with Profile and a different one with Location.

And, yes, when something does not feel right or natural, it may well be a sign that one needs to put more effort in order to understand the pertinent data. In this manner, the Address entity is one of the things that I consider that needs more attention, since I think that the relationship between a Profile and an Address could be handled by means of the Location entity (due to the fact that every Location must have at least one physical Address), therefore we could dismiss the ProfileAddress associative entity depicted in the latest model, but you should continue analizing these points and let me know your ideas.

Also, is it common practice in IDEF1X to change PK/FK denotions in entities for better readability [e.g ProfileId - LocationOwnerProfileId]?

Yes, that is a very clever remark from you, since IDEF1X recommends the use of role names for denominating FOREIGN KEYS, in order to capture the meaning of such attributes in accordance to the entity in which it is being used. It is also worth noting that this is also strongly related to the concept of primary keys migration. As a matter of fact, the use of role names precedes IDEF1X, since it was originally presented by Dr. E. F. Codd in his 1970 seminal text. In this manner, one can clearly see the fidelity that the IDEF1X standard keeps towards the relational model.

I'd be intrigued to learn what you don't particularly like/feel it doesn't model, with/in the solution?

Besides the details already described above about the Address entity, I am not sure if the Roles carried out by a given Profile in a particular Location are equivalent for an Organization or an Person. From my perspective, a Person first needs to be associated with an Organization, and then this Organization would appoint said Person to perform a Role in a particular Location, but you know the scenario better, so this rules may be needless. In this regard, I am going to insist about the fact that it would be very helpful for me to know the contextual description or meaning that the future users of this data structure give to Organization, Profile, and Location, but I understand that this may be considered confidential information, so this would be a limitation.

With the current structure, it seems like everyone (Organization or Person) can be related to anyone (again, Organization or Person) and can be/do anything (Role) anywhere (Location) but, perharps, this is preciselly what you and the users are expecting from this database, for which you will provide well defined constraints, of course. If this is the case, then we are almost providing a final solution. Since, naturally, your opinion is decisive in this situation, you should also analize this ideas and then let me know your conclusions so that we can take the final steps.

Feasible second advance

Unfortunately, the comunication stopped a few weeks ago, I guess because of work commitments that you must meet, which is completely reasonable. I would have been much more content if we had developed a more stable and robust model but, due to our previous interactions, I can assume that I have been able to point you in the right direction.

In addition to what has already been presented in this question and answer process, I consider that providing a new progression from the previous data models may be helpful for other seekers with a similar problem. So, I have created the…

Organizations and Profiles Preliminary Data Model — Second Advance

As can be seen in such data model, I have removed the many-to-many relationship that I have depicted in the preceding models between Profile and Address, since a given Profile is already related to one-to-many Addresses via its owned Locations.

Another change that is illustrated in this new advance is the fact that it now includes the possibility that a given Location can be owned by one-to-many Profiles. Consequently, I have changed the Location PRIMARY KEY (by dismissing the LocationOwnerProfileId attribute) and then added an associative entity (many-to-many) that relates Profile with Location.

Notes

_{1. IDEF1X is a highly recommendable data modeling technique that was defined as a standard in december 1993 by the U.S. National Institute of Standards and Technology (NIST).}

_{2. This is an ocurrence of a (super)type-subtype cluster. In case you are interested, here is an answer in which I deal in a more detailed manner with this kind of relationships.}

_{3. An example of a many-to-many hierarchical relationship, and is very simliar to the structure that gave definitive solution to the “Parts Explotion Problem”. Such solution was, of course, introduced by Dr. Edgar Frank Codd in his 1970 enormously influential paper “A Relational Model of Data for Large Shared Data Banks”.}

_{4. As such, this is an instance of a one-to-many (or many-to-one) hierarchical relationship.}