Empirical networks are rarely binary - capturing strength or frequency of interactions or perhaps even expression or activity of the nodes themselves. This presents a challenge when computing graph measures which require the graph to be unweighted. However, there is a natural expansion of a weighted graph, called a

**filtration**, which allows us to compute our favorite binary graph measures on a weighted graph in a meaningful way.

A filtration is a sequence of binary graphs, each a subgraph of the next. The edge weights of the original weighted network indicate relation strength between nodes, so we construct our filtration with this is mind. The first binary graph in the filtration G_0 includes only the nodes, and subsequent binary graphs G_1, G_2, ... G_N contain the edges of the previous graph as well as some (0 or more) new edges. If we assume all edge weights are unique, we can imagine this process as adding edges to an empty graph one at a time in order of decreasing edge weight.

Then, at each point in the filtration we can compute your favorite graph measure. Now your favorite graph measure is not a number but a function of the edge density.

Then, at each point in the filtration we can compute your favorite graph measure. Now your favorite graph measure is not a number but a function of the edge density.

As mentioned, perhaps it is instead the nodes that have an ordering and the edges do not. The node ordering could be derived from decreasing weights on the nodes such as strength, centrality, or even protein expression for an empirical network, time when a node joins a cluster of interest, etc. Then our filtration moves along the nodes, instead of the edges. However, note the definition of a graph G(V,E) requires edges to connect two nodes that are in the vertex set V, so when a node is added, all edges between this new node and nodes already in the graph are added. Then each new G_i in our filtration contains all the edges and nodes of the previous G_{i-1} in addition to some (0 or more) new nodes and their carried edges. Then we can again compute your favorite graph measure at each point in the filtration to obtain a function on the number of nodes of a weighted node network.