Network graph-theoretical approaches are located half-way between bottom-up and top-down approaches focusing on the relation between the elements of the studied phenomenon. We can roughly describe the network approach as the answer to the question “What can we derive from the sole knowledge of the wiring diagram of a system?”

The classic Konigsberg bridge problem introduced graph theory in 18th century. The problem had the following formulation: does there exist a walk crossing each of the seven bridges of Konigsberg exactly once? The solution to this problem appeared in “Solutio Problematis ad geometriam situs pertinentis” by Eulero (1741). This structure was called a graph and this was the first time a problem was codified in terms of nodes and edges (Figure 1).

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Eulero proved that the problem has no solution. The interest of the Eulero demonstration lies in the fact he considered as the only important feature for the solution the sequence of bridges crossed. He formalized the problem in terms of nodes (land masses) and edges (bridges) connecting the nodes. The resulting mathematical structure is called a graph.

More in general, a graph G is a mathematical object used to model complex structures and it is made of a finite set of vertices (or nodes) V and a collection of edges E connecting two vertices (Figure 2).

Graphs can be defined by descriptors located at local (single nodes), global (entire network), and mesoscale (clusters of nodes, optimal paths) levels. Thus we can compute the degree of each node (how many links are attached to a given node) that is a local, microscopic characteristic or we can compute the so called “average shortest path” or “characteristic length” of a graph corresponding to the average length of minimal paths connecting all the node pairs (this corresponds to a mesoscopic feature of the system) or the general connectivity of the network (A global feature; Watts and Strogatz, 1998; Di Paola et al., 2013).

It is important to stress these different views are strictly intermingled among them, given they derive from the same basic representation (the graph) so that any view influences (and in turn is influenced) by all the others. The necessary (and natural) interaction of different level views is called “middle-out” approach to stress the fact the interest is focused on the mesoscopic level, i.e., on the pattern of between elements relation and not on the fundamental features of the constituting elements (Csermely et al., 2005).

The science that was mostly influenced by this “naturally systemic” view is chemistry that uses since decades the most widespread (and effective) graph formalization: the structural formula (Di Paola et al., 2013). An hydrogen atom embedded into methane molecule has different features than the same hydrogen atom of a water molecule: e.g., the hydrogen in the water molecule has a partial positive charge much greater than the methane hydrogen for the greater electronegative character of oxygen with respect to carbon atom. This is a clear example of top-down causation: the properties of the most basic level (atom) depends on the features of the entire system (molecule). At the same time both methane and water molecules derive their features from the constituent atoms (bottom-up causation). Stressing the two “directions of causality” is in any case out-of-scope, because the chemical graph incorporates both into a global systemic reasoning made it possible by the structural formula or chemical graph.

If we shift to more complex formulas of organic molecules we can appreciate the richness of the possibilities offered by this approach by the way thousands of different quantitative features of the molecules can be directly derived from structural formulas so that, strictly speaking, properties like solubility, melting point, molar refractivity, partition coefficients can be considered as graph descriptors (Todeschini and Consonni, 2008).

Organic chemistry is thus a perfect example of the power of middle-out approach: a chemical graph (structural formula) allows for deriving functional properties of the studied object (chemico-physical and reactivity features), in other words, a “semantics” emerges from a purely syntactical approach (molecular graph). In the case of chemistry, the existence of very powerful “network-based-theories” is a crucial ingredient of the success of network approaches. As a matter of fact the so called “energy of a graph” E(G) can be defined on a purely topological basis as the sum of the absolute values of the eigenvalues of its adjacency matrix (de Abreu et al., 2010), theoretical chemists did know since decades (Gunthard and Primas, 1956) that, under some reasonable assumptions the graph energy was coincident with the total Π-electrons energy according to Huckel molecular orbital theory.

The marked superiority of the simple consideration of molecular graphs over more sophisticated methods in the prediction of biological activity of drugs is an important proof of the power of the graph-theoretical approach in medicinal chemistry (Bender and Glen, 2005).

Is it possible, at least in perspective, to apply this style of reasoning when going into the more “fuzzy” biological world? The work of Denis Noble about the absence of a single privileged level of causation (Noble, 2012) and the need to go from a molecular to a modular approach set forth by Hartwell et al. (1999) go along this direction. In the following we will sketch some operative examples of this kind of approach, focusing on the use of graph-theoretical methods.

A very simple biological proof of the efficiency of the “network-style” of reasoning, beside the much more sophisticated and deep Dehmer et al. (2013) analysis quoted in the introduction, is the prediction of lethal mutants in yeasts by the graph analysis of their metabolic network (Palumbo et al., 2005, 2007).

As we stated before, all the properties relative to each node (edge) must be derived only by its pattern of relations and thus by its peculiar location in the complete graph. In (Palumbo et al., 2005, 2007) the authors checked for the possibility to derive, from purely topological information on the metabolic network of yeast (Saccharomyces cerevisiae), the lethal character of genetic mutations. A metabolic network can be considered as a graph having enzymatic reactions as edges and metabolites as nodes. Since an enzymatic reaction is catalyzed by one or more enzymes, an edge can also represent the enzymes involved in the reaction. The experimental knock-out of an enzyme corresponds to the elimination from the network of the edge (or edges since the same enzyme can catalyze different reactions) corresponding to that particular enzyme (Palumbo et al., 2005). If it is possible to pick up a connectivity descriptor able to unequivocally define essential enzymes (those enzymes whose lack provoke the yeast death) we can safely assume the biological relevance of the metabolism “wiring structure”, irrespective of the specific nature of the involved enzymes.

In the case of yeast metabolic network, the analysis of 36 lethal mutations out of the 412 relative to enzymes involved in metabolism, reported in the Stanford repository (http://www-sequence.stanford.edu/group/yeast_deletion_project/deletions3.html) and in Jeong et al. (2003) and cured by Ma et al. (2004), allowed the authors to discover that the enzymes corresponding to lethal mutations, when deleted, prevent the connections between the separate nodes (Palumbo et al., 2005, 2007). No alternative path is available to connect the separate nodes and this mesoscopic feature based on paths along the network explains the essential character of each specific mutation on a pure topological basis.

This “essentiality-by-location” mesoscopic principle equating the lethal character of a mutation to the lack of an alternative path in the network, was confirmed in (Palumbo et al., 2007) demonstrating that a double mutation involving two enzymes that per se are not essential acquires essentiality and then causes the death of the organism, if the double knock-out provokes the “lack of alternative path” condition. The arising of lethality by the summation of two non-lethal events derives from the existence of a global metabolism architecture and thus cannot be inferred by going in depth into the nature of the two enzymes, in other words is a collective emergent property of the network system (Mikulecki, 2001; Giuliani, 2010).

The lack of exceptions to the “lack of alternative path rule” seems to rule out the existence of lethal mutations deriving by poor kinetics, but this is might be a too strong deduction. Indeed it is worth noting that the analyzed Stanford repository refers to experiments carried out in the same experimental conditions and thus eliminating all the “real life” well known difficulties that make the phenotypic effects of a given mutation strongly context dependent so that even relatively minor variations of nutrients, pH, temperature can exert dramatic effects.

The reported study must be intended as a proof-of-concept of the possibility to observe the “pure topology” properties devised by Tellegen theorem in a biological context, in any case by no means can we consider strictly topological approach as the obliged way for middle-out approaches in biology.

The contemporary presence of “hard wired” topology driven (presence/absence of a link irrespective of its strength is the main driver) and “transient functional” (kinetics plays a relevant role) relations in biological regulation is at the basis of another very interesting application of the “Middle-Out” (starting from the relations) approach.

In their very interesting work, Han et al. (2004) modeled the correlation dynamics of the mutual relation between hubs (proteins engaging a very high number of relations with other proteins, i.e., network elements with a very high node degree) and their partners by using messenger RNA expression profiles.

The authors examined the extent to which hubs in the yeast interactome are co-expressed with their interaction partners: for each hub they computed the average Pearson correlation coefficient (APCC) between the hub mRNA expression and its neighbors and found that APCC followed a bimodal distribution clearly evidencing two distinct hub populations. They called “party hubs” those nodes that were highly correlated as for expression with their partners (high values of APCC). By contrast, they called “date hubs” these nodes characterized by lower APCC values. Looking at particular protein-protein interactions of “party” and “date” hubs the authors discovered high APCC correspond to permanent interactions while “date” hubs correspond to transient interactions (see also Blasi et al., 2005). Moreover, the authors (Han et al., 2004) demonstrated a link between this hub classification and the network tolerance against node breakdown. Scale-free networks are particularly resistant to random node removal (failure) but are extremely sensitive to targeted removal of hubs (attack; Jeong et al., 2001). Han et al. (2004) showed that the removal of party hubs did not affect the network characteristic path length (and consequently the efficiency of network integration), as it happens in the case of failures: conversely when deleting date hubs, the effects were similar to those expected in the case of targeted attacks with a dramatic increase in characteristic length and thus a neat decrease in network communication efficiency.